The delta shock wave for a simplified chromatography system is obtained in the Riemann solution when

The theory of nonlinear chromatography system is the foundation of chromatography separation process and also plays an important role in the modern industry. Recently, the delta shock wave has been captured numerically and experimentally by Mazzotti et al. [

In [

In the present paper, we extend the result from the critical situation

Similar to [

In addition, we emphasize that our result here is also reasonable in the sense of chromatography engineer since a simplified chromatography system can also be derived from the other nonlinear chromatography system:

The system (

The self-similar viscosity vanishing approach was first proposed by Dafermos [

There exist numerous excellent papers for the related equations and results about the measure-valued solutions for hyperbolic systems of conservation laws. The well-known examples are the transport equations [

The paper is organized as follows. In Section

In this section, we generalize the Riemann problem for (

The Riemann solutions to (

The characteristic eigenvalues of (

Now we are in a position to deliver the Riemann solutions to (

If

If

If

If

If

For the case

Let us suppose that

Let

With the above definition, similar to that in [

For the case

Let us check that the

Actually, we can verify the first equation in (

Consequently, for the second equation of (

Through the earlier verification, we can conclude that (

If we take the limit

If we take the limit

The Riemann solution to (

In this section, we are only interested in the viscous regularization of delta shock wave solution to the Riemann problem (

Performing the self-similar transformation

By observing (

With the similar deduction and calculation in Theorem 4.1 in [

For each fixed

Now, we turn our attention to the second equation in (

For each fixed

Then, one has

It follows from Theorem

For any

It is easy to take

Because

For any

Now, we try to prove that

Thus, for any

Let

Take any

By employing integration by parts and applying the fact that

Through simplifying (

Let us turn back to calculate

It is clear that

On the other hand, if we assume that

From the above discussion, we can get the relation

For any

The process of this proof is similar to that of Lemma 5.3 in [

Let

To prove it, we should consider the limit behavior of

Then, by (

It follows from

By taking the limit

Combining (

By taking the limits

Finally, the value of

By substituting (

This work is supported by the National Natural Science Foundation of China (11001116, 11271176) and the Project of Shandong Provincial Higher Educational Science and Technology Program (J11LA03, J12LI01) and Shandong Provincial Natural Science Foundation (ZR2010AL012).